Hilbert geometry of polytopes

Hilbert geometry of polytopes

Andreas Bernig

Abstract. It is shown that the Hilbert metric on the interior of a convex poly- tope is bilipschitz to a normed vector space of the same dimension.

Mathematics Subject Classification (2000).53C60, 53A20, 51F99.

Keywords.Hilbert geometry, Hilbert metric, polytopes.

1. Introduction. Given a compact convex set K in a finite-dimensional vector spaceV, the Hilbert metric (also called Hilbert geometry) on intK is defined by

d(x, y) := 1

2|log[a₁, a2, x, y]|, x, y∈intP.

Here a₁, a₂ are the intersections of the line through xand y with the boundary ofK (ifx=y, one setsd(x, y) := 0). Hilbert metrics are examples of projective Finsler metrics, i.e., Finsler metrics such that straight lines are geodesics. Hilbert’s fourth problem was to classify projective Finsler metrics. This problem was solved by Pogorelov [13], see also [1] for a symplectic approach.

In the last few years, there has been a renewed interest in Hilbert geometries and many research papers were published, see [11, 14, 9, 5, 3, 16, 10, 4] to cite just a few of them.

A natural question is to classify Hilbert metrics up to bilipschitz maps or up to quasi-isometries. In this direction, it was shown by Colbois and Verovic [8] that if∂K is C² with positive Gauss curvature, then the Hilbert metric is bilipschitz to then-dimensional hyperbolic metric. In a similar spirit, it was shown in [4] that the volume entropy of ann-dimensional convex body with C^1,1 boundary equals the volume entropy of hyperbolic space, which isn−1. Since the volume entropy

Supported by the Schweizerischer Nationalfonds grants SNF PP002-114715/1 and 200020- 121506/1.

Andreas Bernig, D´epartement de Math´ematiques, Chemin du Mus´ee 23, 1700 Fribourg, Switzerland

e-mail:[email protected] Published in "Archiv der Mathematik 92(4): 314-324, 2009"

which should be cited to refer to this work.

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is an invariant under bilipschitz maps, the Hilbert metric of a body with C^1,1 boundary can not be bilipschitz to a normed vector space.

On the other extreme, Hilbert metrics of polygons are bilipschitz to normed spaces, as was shown by Colbois, Vernicos and Verovic [6]. Elaborating an argument of Foertsch and Karlsson [9], Colbois and Verovic [7] showed that if the Hilbert metric of a compact convex body is quasi-isometric to a normed vector space (in particular, if it is bilipschitz), then the body is a polytope (recall that a polytope is the convex hull of a finite number of points).

This raised the question whether the converse holds in general, i.e., if the Hilbert metric of any polytope is quasi-isometric (or bilipschitz) to a normed vector space.

The aim of this note is to answer this question in the positive.

Theorem 1.1. LetV be ann-dimensional vector space and letP ⊂V be a compact convex polytope which is described as

P ={x∈V :f1(x)≥0, . . . , fm(x)≥0},

where f1, . . . , fm are affine functions. Let d be the Finsler metric on intP and endow the dual spaceV^∗ with some norm. Then the map

Φ : (intP, d)→V^∗ x→

m i=1

logfi(x)dfi

is a bilipschitz diffeomorphism.

Note that all norms on V^∗ are equivalent, hence we may choose in the proof a Euclidean scalar product and identify V^∗ with V. The map Φ can then be written as

Φ(x) = m

i=1

logf_i(x) gradf_i, where gradfi∈V is the gradient off.

It is an interesting fact (which was communicated to me by J. Lagarias), that a similar map was studied in [12] in connection with trajectories for Karmarkar’s linear programming algorithm.

2. Lipschitz continuity of Φ. Let us fix some notation. The Euclidean norm of a vector inV will be denoted by · 2, and · stands for the Finsler norm on intP.

Forx∈intP and 0=w∈TxintP, we have w=1

2 1

t₁ + 1 t₂

, (1)

wheret₁, t₂>0 andx+t₁w, x−t₂w∈∂P.

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We set R:= max_igradfi₂. By D we will denote the (Euclidean) diameter ofP. We set

P_i:={x∈V :f_i(x) = 0}, which is an affine hyperplane.

Lemma 2.1. i) For allx, y∈intP we have d(x, y)≥ log 2

D x−y2. In particular, forx∈intP andw∈TxintP

w ≥ log 2 D w2.

ii) Let K ⊂intP be a compact set. There exists a constant CK >0 such that for allx∈K and allw∈TxintP

w2≥CKw.

Proof. This follows easily from the definition ofdand from (1).

Lemma 2.2. The mapΦis Lipschitz continuous.

Proof. Letx =y ∈ intP and a₁, a₂ ∈∂P be the intersection points of the line through x and y with ∂P. Without loss of generality, let us assume that a₁ ∈ P₁, a₂∈P₂ and thatxlies betweena₁ andy.

The Hilbert distance betweenxandy is given by d(x, y) = 1

2logy−a12

x−a1₂

x−a22

y−a2₂

= 1

2logf1(y) f1(x)+1

2logf2(x) f2(y)

≥ 1

2max

logf1(x) f1(y)

,

logf2(x) f2(y)

.

Since a1, a2 are the first intersection points with the boundary of P, for each i= 1,2 we have

logf_i(x) fi(y)

≤max

logf₁(x) f1(y)

,

logf₂(x) f2(y)

. It follows from these two inequalities that

d(x, y)≥1

2max

logfi(x) f_i(y)

:i= 1, . . . , m

.

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We now compute Φ(x)−Φ(y)2=

m i=1

logfi(x) gradfi− m i=1

logfi(y) gradfi

2

≤^m

i=1

|logf_i(x)−logf_i(y)| · gradf_i2

≤2mRd(x, y).

Lemma 2.3. Φ is injective.

Proof. Ifx=y, there exists somejwithfj(x)=fj(y). Noting that log is a strictly monotone function, we compute

Φ(x)−Φ(y), x−y= m i=1

(logfi(x)−logfi(y))(fi(x)−fi(y))>0.

The injectivity of Φ follows.

Lemma 2.4. There exists a constant C1 >0 such that for all x ∈ intP and all w∈TxintP

dΦ(w)2≥C1w2.

Proof. The quadratic form w→ ^m_i=1gradfi, w² is positive definite, since the vectors gradfi spanV. Hence there is some constant c1 with

m i=1

gradfi, w²≥c1w²₂.

SinceP is compact, there exists some real numberM >0 withfi(x)≤M for alli and allx∈intP. Therefore,

dΦ(w)₂· w₂≥ dΦ(w), w= m i=1

gradfi, w² fi(x) ≥ c1

Mw²₂.

This proves the lemma, withC1=_M^c1.

3. Lipschitz continuity ofΦ⁻¹. For a fixed polytopeP, we consider two statements (A) and (B).

(A) There exists a constantC >0 such that for allx∈intP and allv∈TxintP we havedΦ(v)2≥Cv.

(B) The map Φ : intP →V is onto and bilipschitz.

Proposition 3.1. If P satisfies (A), then it also satisfies (B).

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Proof. SetW := Φ(intP) =∅. By Lemma 2.4 and the open mapping theorem, W is open. Let c : [0,1] →W be a rectifiable curve and let ˜c := Φ⁻¹◦c be its preimage under Φ. Then

l(c) = 1 0

c(s)2ds= 1 0

dΦ(˜c(s))2ds≥C 1 0

˜c(s)ds=Cl(˜c).

(2)

If W is not equal to V, there exists a curvec : [0,1] →W of finite length with c(t) ∈ W for 0 ≤ t < 1 but c(1) ∈/ W. The preimage ˜c of c|_[0,1) under Φ is a rectifiable curve of finite length in (intP, d). Since this space is complete, ˜c can be extended to the whole interval [0,1]. By the Lipschitz property of Φ, it follows thatc(1) = Φ(˜c(1))∈W, a contradiction. Hence Φ is onto.

Taking forcthe segment between two points, (2) implies that Φ⁻¹ is Lipschitz

with Lipschitz constant _C¹.

Proposition 3.2. (A) is satisfied for simple polytopes.

Proof. We claim that there exists a number >0 such that if we set, forx∈intP, I(x) :={i∈ {1, . . . , m}:fi(x)< },

then

i∈I(x)

Pi=∅ ∀x∈intP.

If no suchexists, we find a zero sequence (l) and pointsxl∈intP with

i∈I_l(xl)

Pi=∅.

Passing to a subsequence, we may assume that the sets I_l(x_l) are all equal to someIand that the sequencex_lconverges to some pointx∈P. Sincef_i(x_l)→0, x∈ ∩_i∈IP_i, which is a contradiction.

For eachx∈intP, the vectors {gradfi :i∈I(x)} are linearily independent.

Indeed, ifpis a vertex of the face∩_i∈I(x)Pi, then the vectors{gradfj:fj(p) = 0}

spanV (see [2]). On the other hand, since P is simple, there are exactly nsuch vectors and hence they are linearily independent.

Let CP > 0 be such that whenever {gradfi, i ∈ I} is some subset of {gradf₁, . . . ,gradf_m} of linearily independent vectors, then for all λ_i ∈ R we

have

i∈I

λ_igradf_i 2

≥C_Pmax

i∈I |λ_i|.

The existence of such a constant follows from the fact that any two norms on a finite-dimensional vector space are equivalent.

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Let x∈ intP and w ∈ TxintP with w₂ = 1. Let t be the real number of minimal absolute value such thatx+tw∈∂P. Thenw ≤ _|t|¹.

We fix a numberδ >0 such that δR≤,mR²

≤CP

2δ.

Let us consider two cases. If|t| ≥δ, then Lemma (2.4) implies that dΦ(w)2≥C1w2=C1≥C1δw.

If|t|< δ, thenx+tw∈Pj for some j and

|gradf_j, w|

fj(x) = 1

|t| >1 δ. It follows thatfj(x)< δR≤, i.e.j∈I(x).

We next compute that dΦ(w)₂=

m i=1

gradfi, w fi(x) gradfi

2

≥

i∈I(x)

gradfi, w f_i(x) gradfi

2

−

i∈I(x)

gradfi, w f_i(x) gradfi

2

≥CP|gradfj, w|

fj(x) −mR²

≥CP

2

|gradfj, w|

f_j(x)

≥CP

2 w.

Property (A) now follows withC:= min

C1δ,^C₂^P

.

Noting that every plane convex polygon is simple, we obtain that Theorem 1.1 holds true in dimension 2. We now proceed by induction on the dimension ofP. We first prove property (A) for polyhedral cones.

In the definition of the Hilbert metric, we supposed K to be compact and convex. However, even ifK is only closed and convex and does not contain any straight line, then the definition makes sense (in this case, one ofa1ora2 may be at infinity).

Proposition 3.3. Let P ⊂V be a polyhedral cone of the form P ={x∈V :f₁(x)≥0, . . . , f_m(x)≥0}

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wheref1, . . . , fm are linear functions onV. Suppose that P does not contain any line, but has non-empty interior. Define a map

Φ : intP →V x→^m

i=1

logfi(x) gradfi.

Then there exists a constant C > 0 such that for each x ∈ intP and each w ∈ TxintP, we have

dΦ(w)2≥Cw.

(3)

Proof. Letu:= ^m_i=1gradfi andE0:=u^⊥. Letx0∈intP and setE:=x0+E0. ThenP^E :=P∩E is a compact polytope of dimensionn−1.

By an easy homogeneity argument, it suffices to prove (3) forx∈intP^E. We letf_i^E denote the restriction of fi to E. With π:V →E0 being the orthogonal projection, we have gradf_i^E=π(gradfi).

SinceP^E:=P∩E is of dimensionn−1, there is a constantC^E such that for allx∈intP^E and allw∈TxintP^E we have

m i=1

gradfi, w

fi(x) gradf_i^E 2

≥C^Ew.

(4)

Fix a positive constantc1 with

(1−c1)u2−mRc1>0.

Let x∈ intP^E and let w ∈ T_xintP be of Euclidean norm 1. We write w = w₁+w₂withw₁parallel toEandw₂parallel to the lineR·x, sayw₂=ρx. Note thatw2=¹₂|ρ|.

By (4),

π(dΦ(w1))=

m i=1

gradfi, w1

f_i(x) gradf_i^E 2

≥C^Ew1.

Next we compute that dΦ(w2) =

m i=1

gradfi, w2

fi(x) gradfi=ρu.

In particular

π(dΦ(w2)) = 0; dΦ(w₂)₂= 2w₂ · u₂. Ifw1 ≥c1w, then we obtain

dΦ(w)₂≥ π(dΦ(w))₂=π(dΦ(w₁))₂≥C^Ew₁ ≥c₁C^Ew.

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If w₁ < c1w, then by triangle inequality w₂ ≥(1−c1)w and, using that Φ is Lipschitz with Lipschitz constant 2mR, whereR= max_igradfi₂(see Lemma 2.2), we get

dΦ(w)₂≥ dΦ(w₂)₂− dΦ(w₁)₂

≥2w2 · u2−2mRw1

≥2(1−c1)u2· w −2mRc1w.

Thus (3) is satisfied withC:= min{c₁C^E,2(1−c1)u₂−2mRc1}.

Proposition 3.4. Property (A) is satisfied for each polytope P of dimensionn.

Proof. Let us prove by induction onk= 0, . . . , n−1 the following statement:

(A_k) For every k-dimensional face F of P, there exists an open neighborhoodU inV and a constantCF such that for allx∈intP∩U and allv ∈TxintP we havedΦ(v)₂≥CFv.

It is clear that (A_n−1), Lemma 2.1 and Lemma 2.4 imply (A). The induction start will be the empty casek=−1.

Suppose nowk≥0. LetF=∩i∈IPibe ak-face ofP. By induction hypothesis, we may assume that there is an open neighborhoodU of thek−1-skeleton ofP and a constantC2 such that for all x ∈intP∩U and allv ∈TxintP we have dΦ(v)2≥C2v.

On the compact set F \U, the continuous functions fj, j /∈ I are strictly positive and hence strictly larger than some constantτ >0. Set

U:={x∈V :fj(x)> τ ∀j /∈I}.

ThenUis open andU∪Uis an open neighborhood ofF.

LetF0be thek-dimensional linear space parallel toF and set ¯V :=V /F0. The affine functionsf_i:V →Rinduce linear functions ¯f_i: ¯V →R. Define a polyhedral cone

P¯ :=

¯x∈V¯ : ¯f_i(¯x)≥0, i∈I .

By Proposition 3.3, applied to ¯P, there exists a constantC₃>0 such that for allx∈intP∩Uand allu∈TxintP∩F₀^⊥ we have

i∈I

u,gradfi fi(x) gradfi

2

≥C3u.

(5)

Hereudenotes the Finsler norm with respect to the polyhedral coneP.

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Letx∈ intP ∩U andw ∈TxintP with w₂ = 1. We write w=w1+w2

withw1∈F₀^⊥ andw2∈F0. Then

dΦ(w2)2=

j /∈I

w2,gradfj fj(x) gradfj

2

≤ mR² τ .

Fix a sufficiently large positive constantc₁with _c ^R

1τ−R <1 and C4:= C3

2

1− R τ c1

−2mR² τ c1 >0.

We consider two cases.

Case 1:w ≤c1. Then dΦ(w)2≥C1w2≥^C_c1¹w.

Case 2:w ≥c1. Lettbe of minimal absolute value such thatx+tw2∈∂P, say x+tw2∈Pj. Since w2 is parallel toF, we havej /∈I. From

τ < fj(x) =|fj(x+tw2)−fj(x)|=|t| · |gradfj, w2| ≤R|t|

we deduce thatw2 ≤ ^R_τ. The triangle inequality now yields w₁ ≥ w − w₂ ≥ w − R

τ c1w ≥ c₁τ−R τ .

Letsbe of minimal absolute value withx+sw1∈∂P, sayx+sw∈Pi. Then

|s| ≤ _w¹₁ ≤ _c1τ−R^τ and thereforefi(x)≤R|s| ≤ _c1^Rτ_τ−R < τ, hencei∈I.

By (1), we have w₁ ≤ _|s|¹. The Finsler norm with respect to the cone ¯P satisfiesw1 ≥ _2|s|¹ . By (5) we get

i∈I

w₁,gradfi fi(x) gradfi

2

≥ C3

2 w₁.

Finally, we obtain

dΦ(w)2≥ dΦ(w1)2− dΦ(w2)2

≥

i∈I

w₁,gradfi fi(x) gradfi

2

−

j /∈I

w₁,gradfj fj(x) gradfj

2

−mR² τ

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≥ C3

2 w1 −2mR² τ

≥ C3

2

1− R τ c1

w − 2mR² τ

=C4w+2mR²

τ c1 w −2mR² τ

≥C4w.

We infer that A_k holds true with

U :=U∪U; C_F := min

C₂,C₁ c1, C₄

.

Theorem 1.1 clearly follows from Propositions 3.4 and 3.1.

Acknowledgements. We wish to thank Gautier Berck for pointing out several er- rors in an earlier version of this manuscript and Bruno Colbois and Patrick Verovic for useful discussions.

Note. The main result of this manuscript, namely that the Hilbert geometry of a polytope is bilipschitz equivalent to a normed space, was also shown independently and with a different proof by Constantin Vernicos [16]

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